184_notes:examples:week4_charge_ring

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184_notes:examples:week4_charge_ring [2018/06/05 14:24] – curdemma184_notes:examples:week4_charge_ring [2021/05/25 14:38] (current) – schram45
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 We begin with an approximation, which will make our calculations simpler, and makes sense based on our representation: We begin with an approximation, which will make our calculations simpler, and makes sense based on our representation:
   * The thickness of the ring is infinitesimally small, and we can approximate it as a circle.   * The thickness of the ring is infinitesimally small, and we can approximate it as a circle.
 +  * The ring is in a perfect circle.
 </WRAP> </WRAP>
 We also make a plan to tackle the integrating, which is a little tougher in cylindrical coordinates. We also make a plan to tackle the integrating, which is a little tougher in cylindrical coordinates.
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 So the dQ in our representation takes up a small angle out of the whole circle, which we can call dϕ. The length of our dQ is therefore Rdϕ (which comes from the [[https://en.wikipedia.org/wiki/Arc_length|arc length]] formula). Remember from the notes on [[:184_notes:dq#dQ_-_Chunks_of_Charge|line charges]] that we can write dQ=λdl. Since the charge Q is uniformly distributed on the ring, we use the length of the ring or circumference (2πR) to write the line charge density λ=Q/2πR. Now, we have an expression for dQ: So the dQ in our representation takes up a small angle out of the whole circle, which we can call dϕ. The length of our dQ is therefore Rdϕ (which comes from the [[https://en.wikipedia.org/wiki/Arc_length|arc length]] formula). Remember from the notes on [[:184_notes:dq#dQ_-_Chunks_of_Charge|line charges]] that we can write dQ=λdl. Since the charge Q is uniformly distributed on the ring, we use the length of the ring or circumference (2πR) to write the line charge density λ=Q/2πR. Now, we have an expression for dQ:
 dQ=λdl=Q2πRRdϕ=Qdϕ2π
 dQ=λdl=Q2πRRdϕ=Qdϕ2π
 +
 +<WRAP TIP>
 +===Assumption===
 +The charge is evenly distributed along the ring. This also assumes the ring is a perfect conductor where charges will distribute evenly along the conductor. If this were not true, the charge density along the ring would not be constant.
 +</WRAP>
  
 To find an expression for →r, we can also consult the representation. →r points from the location of dQ to the point P. The location of dQ is →rdQ=Rˆs. This unit vector ˆs may be unfamiliar, since we are used to working in Cartesian coordinates. ˆs is the unit vector that points along the radius of a cylinder centered on the z-axis in our cylindrical coordinate system. In fact, ˆs actually depends on ϕ, and is more appropriately written as a function in terms of ϕ, or ˆs(ϕ). We do not acknowledge the ϕ-dependence in some of our expressions here, because as you will soon see, all terms containing ˆs will disappear. To find an expression for →r, we can also consult the representation. →r points from the location of dQ to the point P. The location of dQ is →rdQ=Rˆs. This unit vector ˆs may be unfamiliar, since we are used to working in Cartesian coordinates. ˆs is the unit vector that points along the radius of a cylinder centered on the z-axis in our cylindrical coordinate system. In fact, ˆs actually depends on ϕ, and is more appropriately written as a function in terms of ϕ, or ˆs(ϕ). We do not acknowledge the ϕ-dependence in some of our expressions here, because as you will soon see, all terms containing ˆs will disappear.
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